Matakuliah : D 0094 Ekonomi Teknik
Tahun : 2007
Penggunaan ROR dan B/R
Pertemuan 17 dan 18
Contoh Soal Rate of Return:
Contoh :
Vincent Gogh’s painting “Irises”
• John Whitney Payson membeli barang seni ini
seharga $80,000.
• John menjual kembali seharga $53.9 juta 40
tahun kemudian.
• Berapa rate of return dari investasi John ?
Bina Nusantara
Rate of Return
$53.9 Juta
• Diketahui: P =$80,000, F =
$53.9juta, and N = 40 tahun
• Cari : i
• Solusi:
0
40
$53.9 juta  $90.0001  i 
i  17.69%
40
Bina Nusantara
$80,000
Arti Rate of Return
Pada 1970, ketika Wal-Mart Stores, Inc.
menjadi Perusahaan terbuka (go public),
suatu investasi dari 100 saham seharga
$1,650. Investasi tersebut akan menjadi
$13,312,000 pada 31 January 2000.
Berapa rate of return pada investasi
tersebut ?
Bina Nusantara
Solution:
$13,312,000
0
30
$1,650
Diketahui: P = $1,650
F = $13,312,000
N = 30
Cari i:
F  P(1  i )
N
$13,312,000 = $1,650 (1 + i )30
Rate of Return
i = 34.97%
Seandainya anda menginvestasikan senilai ($1,650)
dalam rekening tabungan 6% per tahun. Kemudian
anda hanya memperoleh $9,477 pada January,
2000.
Apa arti dari 6% interest disini?
Ini adalah opportunity cost jika menyimpan uang
pada rekening tabungan, dan merupakan suatu
tindakan yang terbaik yang dapat dilakukan saat itu.
Bina Nusantara
Kemudian, pada tahun 1970, anda memperoleh tawaran
untuk investasi lain dengan interest lebih dari 6% untuk
investasi lain, anda akan mengambil investasi tersebut.
Oleh karena itu, 6% dipandang sebagai minimum
attractive rate of return (rate of return yang dibutuhkan).
Maka, anda dapat menetapkan aturan keputusan berikut
untuk memperoleh investasi yang diusulkan adalah
yang terbaik :
ROR > MARR
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Rate of Return
• Definisi 1: Rate of return (ROR) merupakan
interest rate yang diperoleh pada
unpaid balance dari angsuran suatu
pinjaman.
• Contoh: Sebuah bank meminjamkan $10,000
dan menerima pembayaran pertahun
sebesar of $4,021 selama 3 tahun.
Bank dikatakan memperoleh
penghasilan kembali (return of) 10%
dari pinjaman sebesar $10,000.
Bina Nusantara
Loan Balance Calculation:
A = $10,000 (A/P, 10%, 3)
= $4,021
Year
0
1
2
3
Unpaid
balance
at beg.
of year
-$10,000
-$10,000
-$6,979
-$3,656
Return on
unpaid
balance
(10%)
-$1,000
-$698
-$366
Payment
received
Unpaid
balance
at the end
of year
+$4,021
+$4,021
+$4,021
-$10,000
-$6,979
-$3,656
0
A return of 10% on the amount still outstanding at the
beginning of each year
Bina Nusantara
Rate of Return:
• Definition 2: Rate of return (ROR) is defined
as break-even interest rate, i*, which equates the
present worth of a project’s cash outflows to the
present worth of its cash inflows.
• Mathematical Relation:
PW (i* )  PW (i* )cash inflows  PW (i* )cash outflows
0
Bina Nusantara
Return on Invested Capital
• Definition 3: Return on invested capital is
defined as the interest rate earned on the
unrecovered project balance of an investment
project. It is commonly known as internal rate
of return (IRR).
• Example: A company invests $10,000 in a
computer and results in equivalent annual labor
savings of $4,021 over 3 years. The company is
said to earn a return of 10% on its investment of
$10,000.
Bina Nusantara
Project Balance Calculation:
0
1
2
3
Beginning
project balance
-$10,000
-$6,979
-$3,656
Return on
invested capital
-$1,000
-$697
-$365
-$10,000
+$4,021
+$4,021
+$4,021
Ending project
-$10,000
balance
-$6,979
-$3,656
0
Payment
received
The firm earns a 10% rate of return on funds that remain internally
invested in the project. Since the return is internal to the project, we
call it internal rate of return.
Period
(N)
Project A
Project B
Project C
0
-$1,000
-$1,000
+$1,000
1
-500
3,900
-450
2
800
-5,030
-450
3
1,500
2,145
-450
4
2,000
Project A is a simple investment.
Project B is a nonsimple investment.
Project C is a simple borrowing.
Bina Nusantara
Computational Methods
n
Bina Nusantara
Direct
Solution
Direct
Solution
Log
Quadratic
Project A
Project B
Trial &
Error
Method
Computer
Solution
Method
Project C
Project D
0
-$1,000
-$2,000
-$75,000
-$10,000
1
0
1,300
24,400
20,000
2
0
1,500
27,340
20,000
3
0
55,760
25,000
4
1,500
Direct Solution Methods
• Project A
$1,000  $1,500( P / F , i ,4)
$1,000  $1,500(1  i ) 4
0.6667  (1  i ) 4
ln 0.6667
 ln(1  i )
4
0.101365  ln(1  i )
e
0.101365
 1 i
i  e 0.101365  1
 10.67%
Bina Nusantara
• Project B
PW (i )  $2,000 
$1,300 $1,500

0
2
(1  i ) (1  i )
1
, then
1 i
PW (i )  2,000  1,300 x  1,500 x 2
Let x 
Solve for x:
x  0.8 or -1.667
Solving for i yields
1
1
 i  25%,  1667
.

 i  160%
1 i
1 i
Since  100%  i  , the project's i *  25%.
0.8 
Trial and Error Method – Project C
•
•
Step 1: Guess an interest
rate, say, i = 15%
Step 2: Compute PW(i)
at the guessed i value.
PW (15%) = $3,553
•
Step 3: If PW(i) > 0, then
increase i. If PW(i) <
0,
then decrease i.
PW(18%) = -$749
•
Step 4: If you bracket the
solution, you use a linear
interpolation to
approximate
the solution
3,553
0
-749
15%
18%
 3,553 
i  15%  3%

 3,553  749 
 17.45%
Bina Nusantara
i
Graphical Solution – Project D
• Step 1: Create a NPW plot
using Excel.
• Step 2: Identify the point at
which the curve crosses the
horizontal axis closely
approximates the i*.
• Note: This method is
particularly useful for
projects with multiple rates
of return, as most financial
softwares would fail to find
all the multiple i*s.
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Aturan Keputusan Dasar
If ROR > MARR, terima
Aturan ini tidak berlaku untuk situasi dimana
invesrasi mempunyai ‘multiple rates of
return’
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Problem Multiple Rates of Return
$2,300
$1,000
$1,320
• Cari rate(s) of return:
$2,300 $1,320
PW (i)  $1,000 

1 i
(1  i )2
0
1
Let x 
. Then,
1 i
$2,300 $1,320
PW (i )  $1,000 

(1  i )
(1  i ) 2
 $1,000  $2,300 x  $1,320 x 2
0
Solving for x yields,
x  10 / 11 or x  10 / 12
Solving for i yields
i  10% or 20%
Bina Nusantara
NPW Plot for a Nonsimple Investment with Multiple
Rates of Return
Bina Nusantara
Kalkulasi Kesetimbangan Proyek
i* =20%
n=0
n=1
n=2
+$1,100
+$220
-$1,320
$0
Beg. Balance
Interest
Payment
-$1,000
-$1,000
-$200
+$2,300
Ending Balance
-$1,000
+$1,100
Cash borrowed (released) from the project is assumed to
earn the same interest rate through external investment
as money that remains internally invested.
Critical Issue: Can the company be able to invest the money
released from the project at 20% externally in
Period 1?
If your MARR is exactly 20%, the answer is “yes”, because it
represents the rate at which the firm can always invest
the money in its investment pool. Then, the 20% is also true
IRR for the project.
.
Suppose your MARR is 15% instead of 20%. The assumption
used in calculating i* is no longer valid.
Therefore, neither 10% nor 20% is a true IRR.
Bina Nusantara
How to Proceed: If you encounter multiple
rates of return, abandon the IRR analysis
and use the NPW criterion (or use the procedures
outlined in Appendix A).
• If NPW criterion is used at MARR = 15%
PW(15%) =
-$1,000
+ $2,300 (P/F, 15%, 1)
- $1,320 (P/F, 15%, 2 )
= $1.89 > 0
Investasi diterima
Bina Nusantara
Aturan Keputusan Untuk Nonsimple Investment
•
•
Kemungkinan multiple RORs.
Jika PW (i) plot looks seperti
berikut, maka, IRR = ROR.
i*
i
If IRR > MARR, Terima
Jika PW(i) plot seperti berikut,
maka, IRR  ROR (i*).
• Dapatkan IRR sebenarnya
atau
• Gunakan method PW
PW (i)
•
i*
i*
Bina Nusantara
i
Membandingkan Mutually Exclusive
Alternatives Berdasarkan IRR
• Issue: Dapatkah membuat ranking
‘mutually exclusive projects’ dengan
besaran
dari
IRR?
n
A1
A2
0
-$1,000
-$5,000
1
$2,000
$7,000
IRR
100%
>
40%
$818
<
$1,364
PW (10%)
Incremental Investment
n
0
1
ROR
PW(10%)
Project A1
-$1,000
$2,000
100%
$818
Project A2
-$5,000
$7,000
40%
$1,364
Incremental
Investment
(A2 – A1)
-$4,000
$5,000
25%
$546
• Assuming MARR of 10%, you can always earn that rate from other
investment source, i.e., $4,400 at the end of one year for $4,000
investment.
• By investing the additional $4,000 in A2, you would make additional
$5,000, which is equivalent to earning at the rate of 25%. Therefore,
the incremental investment in A2 is justified.
Incremental Analysis (Procedure)
Step 1:
Step 2:
Step 3:
Compute the cash flows for the difference
between the projects (A,B) by subtracting
the cash flows for the lower investment
cost project (A) from those of the higher
investment cost project (B).
Compute the IRR on this incremental
investment (IRR B-A ).
Accept the investment B if and only if
IRR B-A > MARR
Bina Nusantara
Contoh - Incremental Rate of Return
n
B1
B2
0
1
2
3
-$3,000
1,350
1,800
1,500
-$12,000
4,200
6,225
6,330
25%
17.43%
IRR
B2 - B1
-$9,000
2,850
4,425
4,830
15%
Diketahui MARR = 10%, project mana pilihan terbaik?
Bila IRRB2-B1=15% > 10%, dan juga IRRB2 > 10%, pilih B2.
Bina Nusantara
IRR pada Increment Investment:
Tiga Alternatif
n
D1
D2
D3
0
-$2,000
-$1,000
-$3,000
1
1,500
800
1,500
2
1,000
500
2,000
3
800
500
1,000
34.37%
40.76%
24.81%
IRR
Step 1: Tetapkan IRR untuk tiap
proyek untuk mngeliminasi tiap
project yang gagal memenuhi
MARR
Step 2: Bandingkan D1 dan D2 berpasangan.
IRRD1-D2=27.61% > 15%,
pilih D1.
Step 3: Bandingkan D1 dan D3.
IRRD3-D1= 8.8% < 15%,
pilih D1.
Kesimpulannya D1 adalah alternatif terbaik.
Bina Nusantara
Incremental Borrowing Analysis
Principle:
• If the difference in flow (B-A)
represents an increment of
investment, then (A-B) is an
increment of borrowing.
• When considering an
increment of borrowing, the
rate i*A-B is the rate we paid
to borrow money from the
increment.
Bina Nusantara
Decision Rule:
i * A  B  BRR A  B
• If BRR B-A < MARR, select B.
• If BRR B-A = MARR, select
either one.
• If BRR B-A > MARR, select A.
Borrowing Rate of Return
n
B1
0
-$3,000
1
1,350
4,200
-2,850
2
1,800
6,225
-4,425
3
1,500
6,330
-4,830
Bina Nusantara
B2
B1-B2
-$12,000 +$9,000
Incremental Analysis for Cost-Only Projects
Items
CMS Option
FMS Option
Annual O&M costs:
Annual labor cost
$1,169,600
$707,200
832,320
598,400
3,150,000
1,950,000
Annual tooling cost
470,000
300,000
Annual inventory cost
141,000
31,500
1,650,000
1,917,000
Total annual costs
$7,412,920
$5,504,100
Investment
$4,500,000
$12,500,000
$500,000
$1,000,000
Annual material cost
Annual overhead cost
Annual income taxes
Net salvage value
Bina Nusantara
Incremental Cash Flow (FMS – CMS)
n
Bina Nusantara
CMS Option
FMS Option
Incremental
(FMS-CMS)
0
-$4,500,000
-$12,500,000
-$8,000,000
1
-7,412,920
-5,504,100
1,908,820
2
-7,412,920
-5,504,100
1,908,820
3
-7,412,920
-5,504,100
1,908,820
4
-7,412,920
-5,504,100
1,908,820
5
-7,412,920
-5,504,100
1,908,820
6
-7,412,920
-5,504,100
Salvage
+ $500,000
+ $1,000,000
$2,408,820
Solution:
PW (i) FMS CMS  $8,000,000
$1,908,820( P / A, i,5)
$2,408,820( P / F, i,6)
0
IRRFMS CMS  12.43%  15%,
select CMS.
Bina Nusantara
Ultimate Decision Rule:
If IRR > MARR, Accept
• Aturan berlaku untuk setiap situasi investasi.
• Dalam beberapa situasi,
IRR = ROR
tapi hubungan ini tidak dapat digunakan untuk
investasi dengan multiple RORs.
Bina Nusantara
Predicting Multiple RORs
- 100% < i *< infinity
• Net Cash Flow Rule of Signs
No. of real RORs (i*s)
<
No. of sign changes in the project
cash flows
Bina Nusantara
Example
n
0
1
2
3
4
5
6
Net Cash flow
-$100
-$20
$50
0
$60
-$30
$100
Sign Change
1
1
1
• No. of real i*s  3
• This implies that the project could have
(0, 1, 2, or 3) i*s but NOT more than 3.
Bina Nusantara
Accumulated Cash Flow Sign Test
Find the accounting sum of net cash flows at
the end of each period over the life of the
project
Period
Cash Flow
Sum
(n)
(An )
Sn
0
A0
S0  A0
1
A1
S1  S0  A1
2
A2
S2  S1  A2


N
AN

SN  SN 1  AN
If the series S starts negatively and changes sign
ONLY ONCE, there exists a unique positive i*.
Bina Nusantara
Contoh
n
An
Sn
0
1
2
3
4
5
6
-$100
-$20
$50
0
$60
-$30
$100
-$100
-$120
-$70
-$70
-$10
-$40
$60
Perubanhan
tanda
1
• Jumlah tanda berubah = 1, indicating a unique i*.
• i* = 10.46%
Bina Nusantara
$3,900
Contoh:
0
$2,145
2
1
3
$1,000
$5,030
• Apakah ini simple investment?
• Berapa RORs (i*s) dapat diharapkan dari menguji
cash flows?
• Apakah Investasi ini mempunyai rate of return yang unik ?
Bina Nusantara
Contoh Soal-soal
Benefit Cost Ratio
Bina Nusantara
Contoh : Incremental Benefit-Cost Ratios
A1
A3
I
$5,000
$20,000
$14,000
B
12,000
35,000
21,000
C’
4,000
8,000
1,000
$3,000
$7,000
$6,000
PW(i)
Bina Nusantara
A2
Solution
A1
A2
A3
BC(i)
1.33
1.25
1.40
Ranking Base
A1
A3
A2
I +C’
$9,000
$15,000
$28,000
$21,000  $12,000
BC(i)2 1 
($14,000  $5,000)  ($1,000  $4,000)
 1.5  1, select A2.
$35,000  $21,000
BC(i)2 3 
($20,000  $14,000)  ($8,000  $1,000)
 1.08  1, select A2.
General Procedure for Cost-Effectiveness
Studies
Step 1: Establish the goals to be achieved by the analysis.
Step 2: Identify the imposed restrictions on achieving the goals, such
as budget or weight.
Step 3: Identify all the feasible alternatives to achieve the goals.
Step 4: Identify the social interest rate to use in the analysis.
Step 5: Determine the equivalent life-cycle cost of each alternative,
including research and development, testing, capital investment,
annual operating and maintenance costs, and salvage value.
Bina Nusantara
•
Step 6: Determine the basis for developing the cost-effectiveness
index. Two approaches may be used;
– (1) the fixed-cost approach and
– (2) the fixed-effectiveness approach.
– If the fixed-cost approach is used, determine the amount of
effectiveness obtained at a given cost.
– If the fixed-effectiveness approach is used, determine the cost to
obtain the predetermined level of effectiveness.
•
Step 7: Compute the cost-effectiveness ratio for each alternative based
on the selected criterion in Step 6.
•
Step 8: Select the alternative with the maximum cost-effective index.
Bina Nusantara
Cost-Effectiveness Decision Criterion
• Fixed Cost Approach
Maximize Effectiveness
Minimize Cost
Subject to:
Subject to:
Budget Constraint
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• Fixed Effectiveness
Approach
Must meet the minimum
effectiveness
Case Study - Selecting an Weapon
System
Bina Nusantara
Weapon System Alternatives
Alternative Aj
Advantage
Disadvantage
Probability
of Kill
A1: Inertial navigation
system
Low cost, mature
technology.
Accuracy, target
recognition
0.33
A2: Inertial navigation
system: Global
positioning system
Moderate cost, nature
technology
Target recognition
0.70
A3: Imaging infrared
(I2R)
Accurate, target
recognition
High cost, bunkered
target detection
0.90
A4: Synthetic aperture
radar
Accurate, target
recognition
High cost
0.99
A5: Laser
detection/ranging
Accurate, target
recognition
High cost, technical
maturity
0.99
A6: Millimeter wave
(MMW)
Moderate cost,
accurate
Target recognition
0.80
Life-Cycle Costs for Weapon Development
Alternative
Expenditures in Million Dollars
Phase
FSD
IOC
PW(10%)
Year
A1*
A2
A3
A4
A5
A6
0
$15
$19
$50
$40
$75
$28
1
18
23
65
45
75
32
2
19
22
65
45
75
33
3
15
17
50
40
75
27
4
90
140
200
200
300
150
5
95
150
270
250
360
180
6
95
160
280
275
370
200
7
90
150
250
275
340
200
8
80
140
200
200
330
170
$315.92
$492.22
$884.27
$829.64
$1,227.23
$612.70
Cost-Effectiveness Index
Type
Bina Nusantara
Cost/Unit
Probability of
Kill
Cost/Kill
Kill/Cost
A1
$31,592
0.33
$95,733
0.0000104
A2
49,220
0.70
70,314
0.0000142
A3
88,427
0.90
98,252
0.0000102
A4
82,964
0.90
83,802
0.0000119
A5
122,723
0.99
123,963
0.0000081
A6
61,370
0.80
76,713
0.0000130
Unacceptable
region
$130,000
A5
120,000
Cost/kill
110,000
Fixed cost
100,000
A1
A3
90,000
Maximize
effectiveness
A4
80,000
70,000
300
A6
A2
400
500
600
700
800
900
1000 1100
Present value of life cycle cost ($ million)
1200 1300